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A phase field approach to shape optimization in Navier–Stokes flow with integral state constraints

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Abstract

We consider the shape optimization of an object in Navier–Stokes flow by employing a combined phase field and porous medium approach, along with additional perimeter regularization. By considering integral control and state constraints, we extend the results of earlier works concerning the existence of optimal shapes and the derivation of first order optimality conditions. The control variable is a phase field function that prescribes the shape and topology of the object, while the state variables are the velocity and the pressure of the fluid. In our analysis, we cover a multitude of constraints which include constraints on the center of mass, the volume of the fluid region, and the total potential power of the object. Finally, we present numerical results of the optimization problem that is solved using the variable metric projection type (VMPT) method proposed by Blank and Rupprecht, where we consider one example of topology optimization without constraints and one example of maximizing the lift of the object with a state constraint, as well as a comparison with earlier results for the drag minimization.

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Acknowledgments

The authors gratefully acknowledge the financial support by the Deutsche Forschungsgemeinschaft (DFG) through the grants GA695/6-2 (first and fourth author) and HI689/7-1 (second and third author) within the priority program SPP1506 “Transport processes at fluidic interfaces”. The third author additionally gratefully acknowledges the support by the DFG through the International Research Training Group IGDK 1754 “Optimization and Numerical Analysis for Partial Differential Equations with Nonsmooth Structures”.

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Authors and Affiliations

  1. Fakultät für Mathematik, Universität Regensburg, 93040, Regensburg, Germany

    Harald Garcke

  2. Fachbereich Mathematik, Universität Hamburg, Bundesstrasse 55, 20146, Hamburg, Germany

    Michael Hinze

  3. Zentrum Mathematik, Technische Universität München, Garching bei München, Germany

    Christian Kahle

  4. Department of Mathematics, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong

    Kei Fong Lam

Authors
  1. Harald Garcke
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  2. Michael Hinze
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  3. Christian Kahle
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  4. Kei Fong Lam
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Correspondence to Harald Garcke.

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Communicated by: Jon Wilkening

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Garcke, H., Hinze, M., Kahle, C. et al. A phase field approach to shape optimization in Navier–Stokes flow with integral state constraints. Adv Comput Math 44, 1345–1383 (2018). https://doi.org/10.1007/s10444-018-9586-8

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  • DOI: https://doi.org/10.1007/s10444-018-9586-8

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